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Linear Equations and Inequalities

5.1 Rectangular Coordinate System

Graph the ordered pairs (-1,3), (1,4), (-4,0), and (-2,-1).

Find the coordinates of each of the points

Find the coordinates of each of the points

Scatter DiagramsIn March of 1998, the American Council for an Energy-Efficient Economy
released its ranking of environmentally friendly and unfriendly cars and trucks
sold in the United States. The following table shows the fuel use, in miles per
gallon of gasoline, both in the city and on the highway, for six on the vehicles
ranked worst for the environment. Graph the scatter diagram for these data.

City fuel use, x

12

9

10

12

13

12

Highway use, y

16

15

13

16

17

17

5.2 Graphs of Straight Lines

A solution of a linear equation in two variables is an
ordered pair of numbers (x,y) that makes the equation a true statement.

y = mx + b

For y = 3x – 5, find the value of m and the value of b. Is
(-1,-8) a solution?

Is (2, -4) a solution of
?

Find the ordered pair solution of
that corresponds to x = 4.

A graph of an equation in two variables is a drawing of
the ordered pair solutions of the equation.
• find three solutions to the equation
• graph the points
• draw a line through the points

Graph y = 2x + 2

Graph y = -3x + 1

Graph

Graph

Graph y = 3x

Graph equations of the form
Ax + By = C
by first rewriting to the form
y = mx + b
then
• find three solutions to the equation
• graph the points
• draw a line through the points

Write 5x – 2y = 10 in the form y = mx + b then graph the
equation.

Graph x – 3y = 9

x-intercept is where the graph crosses the x-axis, (x,0).
y-intercept is where the graph crosses the y-axis, (0,y).

Find the x- and y-intercepts for 4x – y = 4. Graph the
line.

Graph of y = b

Graph of x = a

Graph y = 3

Graph x = -4

5.3 Slopes of Straight Lines

The slope of a line is the ratio of the change in the
y-coordinates between any two points on the line to the change in x-coordinates.

Slope = m =

Find the slope of the line containing points (-1,1) and
(2,3).

Find the slope of the line containing points (-3,4) and
(2,-2).

Find the slope of the line containing points (-1,3) and
(2,3).

Find the slope of the line containing points (2,4) and
(2,-2).

Find the slope of the line containing points (-1,2) and
(1,3).

Find the slope of the line containing points (1,2) and
(4,-5).

Find the slope of the line containing points (2,3) and
(2,7).

Find the slope of the line containing points (1,-3) and
(-5,-3).

For any equation of the form
y = mx + b,
the slope of the line is m, the y-intercept is b.

Graph y = 2x + 3

Graph

Graph x – 2y = 4

5.4 Equations of Straight Lines

Slope intercept form: y = mx + b

Find the equation of the line that has slope 2 and
y-intercept (0,3).

Find the equation of the line that has slope 3/2 and
contains the point (4,-2).

Point-Slope Formula

y – y1 = m(x – x1)

Use the point slope formula to find the equation of a line
that passed through the point whose coordinates are (-2, -1) and has slope 3/2.

Use the point slope formula to find the equation of a line
that passed through the point whose coordinates are (5,4) and has slope 2/5.

5.5 Functions

Set = a collection of objects

Eg.

Relation = set of ordered pairs

Eg.

Domain of a relation = set of first coordinates of the ordered pairs.

Eg.

Range of a relation = set of second coordinates of the ordered pairs.

Eg.

Function = a relation where no two ordered pairs have the same first coordinate
and different second coordinates.

Eg.

Find the domain and range of the relation {(-5, 1), (-3, 3), (-1, 5)}. Is the
relation a function?

Find the domain and range of the relation {(1,0), (1,1), (1,2), (1,4), (1,4)}.
Is the relation a function?

f(x) is the value of the function at x

f(x) = x2
f( ) = ( )2

Evaluate f(x) = 2x – 4 at x = 3. Write an ordered pair that is an element of the
function.

Evaluate f(x) = -5x + 1 at x = 2. Write an ordered pair that is an element of
the function.

When a function is described by an equation and the domain is specified, the
range of the function can be found.

Find the range of the function given by the equation f(x) = -3x + 2 if the
domain is {-4, -2, 0, 2, 4}

Find the range of the function given by the equation f(x) = 4x - 3 if the domain
is {-5, -3, -1, 1}

Graphs of linear functions

Equations of the form y = mx + b are functions.

y = dependent variable
x = independent variable

Can write y = 2x + 1 in functional notation.

Functions of the form f(x) = mx + b are called linear functions.

Graph

The value, V, of an investment of $2500 at an annual simple interest rate of 6%
is given by the equation V = 150t + 2500, where t is the amount of time, in
years, that the money is invested. Write the equation in functional notation.
Graph the equation for values of t between 0 and 10. The point whose coordinates
are (5,3250) is on the graph. Write a sentence that explains the meaning of this
ordered pair.

A car is traveling at a uniform speed of 40 mph. The distance, d (in miles), the
car travels in t hours is given by the equation d = 40 t. Write the equation in
functional notation. Graph the equation for values of t between 0 and 5. The
point whose coordinates are (3, 120) is on the graph. Write a sentence that
explains the meaning of this ordered pair.

5.6 Graphing Linear Inequalities.

The graph of y = x + 1 separates a plane into three sets:

y = x + 1

y > x + 1

y < x + 1

• Solve for y
• Graph the line
• Decide which half-plane
• Solid line or dashed line